Waves in parallel or swirling stratified shear flows

1979 ◽  
Vol 93 (3) ◽  
pp. 401-412 ◽  
Author(s):  
S. Leibovich
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Wave Energy ◽  
Richardson Number ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Maximum Speed ◽  
Meridian Plane ◽  
Finite Rate ◽  
Critical Richardson Number

The dispersion relations for infinitesimal internal gravity waves (A) and axisymmetric waves in swirling streams (B) are considered. In both cases the mainstream may be sheared and density stratified in the transverse (vertical in case A, radial in case B) direction. The following results are proved for either case: If the maximum speed Wmax (or minimum speed Wmin) (in a meridian plane in case B) of the mainstream occurs at an interior point in the fluid, then the phase speed of any mode takes all values from the Wmax (or Wmin) to +∞ ( —∞) as the overall Richardson number λ2 varies from 0 to ∞. If Wmax(Wmin) is attained at a boundary point with finite rate of strain, there is a positive non-zero critical Richardson number below which one or both branches of the dispersion relation terminate. These results employ variational methods and correct erroneous results concerning problem B stated in Chandrasekhar's treatise on hydrodynamic stability. Furthermore, bounds are given on the group velocity for both branches of the dispersion relation. From these bounds it is shown that in the absence of reversals of the mainstream (Wmin > 0) upstream propagation of wave energy is impossible whenever upstream propagation of constant phase surfaces is impossible.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

scholarly journals Momentum and Energy Transport by Gravity Waves in Stochastically Driven Stratified Flows. Part I: Radiation of Gravity Waves from a Shear Layer

2007 ◽  
Vol 64 (5) ◽  
pp. 1509-1529 ◽  
Author(s):  
Nikolaos A. Bakas ◽  
Petros J. Ioannou
Keyword(s):  
Shear Flow ◽  
Shear Layer ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Wave Energy ◽  
Energy Transport ◽  
Wave Spectrum ◽  
Internal Gravity Waves ◽  
Wave Interference ◽  
Wide Range

Abstract In this paper, the emission of internal gravity waves from a local westerly shear layer is studied. Thermal and/or vorticity forcing of the shear layer with a wide range of frequencies and scales can lead to strong emission of gravity waves in the region exterior to the shear layer. The shear flow not only passively filters and refracts the emitted wave spectrum, but also actively participates in the gravity wave emission in conjunction with the distributed forcing. This interaction leads to enhanced radiated momentum fluxes but more importantly to enhanced gravity wave energy fluxes. This enhanced emission power can be traced to the nonnormal growth of the perturbations in the shear region, that is, to the transfer of the kinetic energy of the mean shear flow to the emitted gravity waves. The emitted wave energy flux increases with shear and can become as large as 30 times greater than the corresponding flux emitted in the absence of a localized shear region. Waves that have horizontal wavelengths larger than the depth of the shear layer radiate easterly momentum away, whereas the shorter waves are trapped in the shear region and deposit their momentum at their critical levels. The observed spectrum, as well as the physical mechanisms influencing the spectrum such as wave interference and Doppler shifting effects, is discussed. While for large Richardson numbers there is equipartition of momentum among a wide range of frequencies, most of the energy is found to be carried by waves having vertical wavelengths in a narrow band around the value of twice the depth of the region. It is shown that the waves that are emitted from the shear region have vertical wavelengths of the size of the shear region.

scholarly journals Dependence of the critical Richardson number on the temperature gradient in the mesosphere

2018 ◽  
Author(s):  
Michael N. Vlasov ◽  
Michael C. Kelley
Keyword(s):  
Temperature Gradient ◽  
Atmospheric Turbulence ◽  
Gravity Waves ◽  
Wind Shear ◽  
Richardson Number ◽  
Buoyancy Force ◽  
Dynamic Instability ◽  
Critical Richardson Number ◽  
Vertical Accelerations ◽  
First Time

Abstract. Maximum upper atmospheric turbulence results in the mesosphere from convective and/or dynamic instabilities induced by gravity waves. For the first time, by comparing the vertical accelerations induced by wind shear and the buoyancy force, it is shown that the critical Richardson number Ric can be estimated. Dynamic instability is developed for Ri 

scholarly journals Parametrization of momentum and energy depositions from gravity waves generated by tropospheric hydrodynamic sources

1997 ◽  
Vol 15 (12) ◽  
pp. 1570-1580 ◽  
Author(s):  
N. M. Gavrilov
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Energy Fluxes ◽  
The Mean ◽  
Turbulence Production

Abstract. The mechanism of generation of internal gravity waves (IGW) by mesoscale turbulence in the troposphere is considered. The equations that describe the generation of waves by hydrodynamic sources of momentum, heat and mass are derived. Calculations of amplitudes, wave energy fluxes, turbulent viscosities, and accelerations of the mean flow caused by IGWs generated in the troposphere are made. A comparison of different mechanisms of turbulence production in the atmosphere by IGWs shows that the nonlinear destruction of a primary IGW into a spectrum of secondary waves may provide additional dissipation of nonsaturated stable waves. The mean wind increases both the effectiveness of generation and dissipation of IGWs propagating in the direction of the wind. Competition of both effects may lead to the dominance of IGWs propagating upstream at long distances from tropospheric wave sources, and to the formation of eastward wave accelerations in summer and westward accelerations in winter near the mesopause.

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

1978 ◽  
Vol 85 (1) ◽  
pp. 7-31 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Shear Flow ◽  
Internal Waves ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Depth Interval ◽  
Plane Shear

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

On the mean motion induced by internal gravity waves

1969 ◽  
Vol 36 (4) ◽  
pp. 785-803 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Wave Train ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Motion ◽  
The Mean

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

scholarly journals Analogy between electro-magnetic waves in cold unmagnetized plasma and shallow water inertio-gravity waves in geophysical systems

2021 ◽  
Author(s):  
E. Heifetz ◽  
L. R. M. Maas ◽  
J. Mak ◽  
I. Pomerantz
Keyword(s):  
Dispersion Relation ◽  
Shallow Water ◽  
Gravity Waves ◽  
Collisionless Plasma ◽  
Water System ◽  
Phase Speed ◽  
Rotating Shallow Water ◽  
Unmagnetized Plasma ◽  
Electro Magnetic ◽  
Magnetic Waves

Abstract The fundamental dispersion relation of transverse electro-magnetic waves in a cold collisionless plasma is formally equivalent to the two-dimensional dispersion relation of inertio-gravity waves in a rotating shallow water system, where the Coriolis frequency can be identified with the plasma frequency, and the shallow water gravity wave phase speed plays the role of the speed of light. Here we examine this equivalence and compare between the propagation wave mechanisms in these seemingly unrelated physical systems.

scholarly journals A Global Model for the Diapycnal Diffusivity Induced by Internal Gravity Waves

2013 ◽  
Vol 43 (8) ◽  
pp. 1759-1779 ◽  
Author(s):  
Dirk Olbers ◽  
Carsten Eden
Keyword(s):  
Energy Density ◽  
Internal Wave ◽  
Wave Field ◽  
Ocean Circulation ◽  
Gravity Waves ◽  
Wave Energy ◽  
Internal Gravity Waves ◽  
Radiation Balance ◽  
Dissipation Rates ◽  
Diapycnal Diffusivity

Abstract An energetically consistent model for the diapycnal diffusivity induced by breaking of internal gravity waves is proposed and tested in local and global settings. The model [Internal Wave Dissipation, Energy and Mixing (IDEMIX)] is based on the spectral radiation balance of the wave field, reduced by integration over the wavenumber space, which yields a set of balances for energy density variables in physical space. A further simplification results in a single partial differential equation for the total energy density of the wave field. The flux of energy to high vertical wavenumbers is parameterized by a functional derived from the wave–wave scattering integral of resonant wave triad interactions, which also forms the basis for estimates of dissipation rates and related diffusivities of ADCP and hydrography fine-structure data. In the current version of IDEMIX, the wave energy is forced by wind-driven near-inertial motions and baroclinic tides, radiating waves from the respective boundary layers at the surface and the bottom into the ocean interior. The model predicts plausible magnitudes and three-dimensional structures of internal wave energy, dissipation rates, and diapycnal diffusivities in rough agreement to observational estimates. IDEMIX is ready for use as a mixing module in ocean circulation models and can be extended with more spectral components.

A note on breaking waves

1988 ◽  
Vol 419 (1857) ◽  
pp. 323-335 ◽  
Keyword(s):  
Phase Velocity ◽  
Gravity Waves ◽  
Turbulent Mixing ◽  
Wave Breaking ◽  
Richardson Number ◽  
Breaking Waves ◽  
Internal Gravity Waves ◽  
Time Interval ◽  
Wave Groups ◽  
Surface Gravity Waves

Some simple general properties of wave breaking are deduced from the known behaviour of surface gravity waves in deep water, on the assumption that breaking occurs in association with wave groups. In particular we derive equations for the time interval, ז, between the onset of breaking of successive waves: ז ═ T / |1 – ( c ⋅ c g )/ c 2 |, and for the propagation vector c b (referred to as the ‘wave-breaking vector’) of the position at which breaking, once initiated, will proceed: c b ═ c (1 – c ⋅ c g / c 2 )+ c g . Here c is the phase velocity, and c g the group velocity, of waves of period T . Interfacial waves, internal gravity waves, inertial waves and planetary waves are considered as particular examples. The results apply not only to wave breaking, but to the movement of any property (e. g. fluid acceleration, gradient Richardson number) that is carried through a medium in association with waves. One application is to describe the distribution, in space and time, of regions of turbulent mixing, or transitional phenomena, in the oceans or atmosphere.

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